1. René Descartes contributed a lot in the field of mathematics including La Géométrie and Le Mond, ou Traité de la Lumiére. Descartes' rule of signs makes it easier for us to figure out how many positive and negative roots there are to a polynomial. Descartes was the founder of Cartesian geometry, which uses algebra to express geometry. 2. I believe that La Géométrie is the most important contribution to mathematics that René Descartes had. I think that algebra and geometry should be connected to each other because algebra makes graphing very easy. Without doing the algebraic work required to get your equation for the graph, it would be a lot harder to graph things.
3. Maria Gaetana Agnesi grew up in a very wealthy lifestyle. I find this extremely interesting because most of the mathematicians in her time period grew up and made a name for themselves. She had great influence from her father to learn many things and teach those to her twenty younger brothers and sisters.
4. The concept that I found most useful from lesson 8-1 was the distance formula. Typically the distance formula is used, on a graph, to find the distance between two points. However, the distance formula is used in our every day lives quite frequently. It is used to find the distance between two cities, which we see on road signs. It is used in football games by saying how many yards it will take for the football to get from the line of scrimage to the first down marker. Some things we use the distance formula for, we don't even realize that we are using it. The distance formula is something that you can use for the rest of your life, evn when you are out of your math class!
5. y = (-1/2)x + 3
April 29, 2011
1. Science and mathematics are so closely related that they could easily be intertwined. Many of the great mathematicians could be great in the science field as well and that could be said for many scientists as well about the field of mathematics. One example of this is Johann Kepler.
2. 3. The most difficult problem that I faced in chapter 8 was in section 8-7( #42 on page 472).
2x - y + 3z = 6
x + 2y - z = 8
2y + z = 1
First, I decided to get rid of the z's so I chose the bottom two equations.
x + 2y - z = 8 2y + z = 1
x + 4y = 9
Then, I chose the top and the bottom (multiplied by -3) to eliminate z as well.
2x - y + 3z = 6 -6y - 3z =-3
2x - 7y = 3
I took my two result equations and eliminated x. to do this I multiplied my first result equation by -2. Then I solved for y.
-2x - 8y = -18 2x - 7y = 3
-13y=-15
y=(15/13)
May 13, 2011
1. The Greeks believed that geometry best expressed mathematical science. The Greeks felt that rulers were inadequate for making constructions and they only allowed two instruments: a compass and a straight edge. The straight edge could have absolutely no markins on it and they could not line up points by eye. There were three constructions that the Greeks could not construct with these tools and are known as the three famous problems of aniquity.
-To trisect an arbitrary angle.
-To construct the length of the edge of a cube having twice the volume of a given cube.
-To construct a square having the same area as that of a given circle.
I find the idea of geometry being such a powerful mathematical represetation of science to be very interesting. I myself have been an algebra person because all you need to know is how to set up a formula and then it's easy to solve it. Lately, however, my feeling toward geometry have changed exponentially and I have had a lot of fun working on constructions in my math class (We did have to use a straight edge with markings on it though). I undertand how the Greeks could feel so strongly toward geometry, and in time I might feel the same way.
2. Geometrical shapes are a large part of our world whether or not we always notice them. They are seen all throughout the day and we just don't think about them. The shape that most people see andthink about is the octagon. The octagon is on the famous 'STOP' sign that some of us refuse to abide by. Imagine how wierd the 'STOP' sign would look if it was a triangle. Speaking of triangles, if you live in a house the chances of your roof being held above your house rather than in it, come from our triangular friends "rafters".
If you have ever ordered a pizza, which I have definately ordered my share, then you know that it comes in a nice square pizza box. Who knew that a square would have such a delicious treat stowed away inside of it? Rectangles are the friends to many sports fans that play and/or watch sports. Many of the sports in the world are played on rectangular fields or courts. The pentagon, afive sided shape is also the headquarters of the United States Department of Defense. The building is actually in the shape of a pentagon.
3. Euclid is known as the "Father of Geometry" because he, like Thales, Pathagoras, Plato and more, found it easier to calculate things with geometry. The Greek number system was not a positional system which means that there were no zero's and there were only whole numbers which made it very difficult to make calculations. Euclid decided to use a compass and a straight edge to do calculations in geometrical form and these techniques later became known as Euclidean Constructions. He was the author The Elements, which covered plane geometry, arithmetic and number theory, irrational numbers and solid geometry.
If i could ask Euclid any question, I would ask him how he proved the Pythagorean Theorem. I have seen many different proofs and I, myself, could solve it in many different ways, but I would love to see Euclid solve it. I would imagine that he would come up with the most complex way to solve it and it would probably take a long time as well, but I would be very pleased to his method of solving the Pythagorean Theorem.
4. By definition, a parallelogram is a quadrilateral, meaning four sides, with two pairs of parallel sides. A rhombus is a parallelogram with all sides having equal length. A rhombus has all the criteria to be considered a parallelogram because it has two pairs of parallel sides. However, a parallelogram does have two pairs of parallel sides, like a rhombus, but not all of the sides are equal. If all of the sides were equal, then it would be a rhombus and a parallelogram.
April 15, 2011
1. René Descartes contributed a lot in the field of mathematics including La Géométrie and Le Mond, ou Traité de la Lumiére. Descartes' rule of signs makes it easier for us to figure out how many positive and negative roots there are to a polynomial. Descartes was the founder of Cartesian geometry, which uses algebra to express geometry.
2. I believe that La Géométrie is the most important contribution to mathematics that René Descartes had. I think that algebra and geometry should be connected to each other because algebra makes graphing very easy. Without doing the algebraic work required to get your equation for the graph, it would be a lot harder to graph things.
3. Maria Gaetana Agnesi grew up in a very wealthy lifestyle. I find this extremely interesting because most of the mathematicians in her time period grew up and made a name for themselves. She had great influence from her father to learn many things and teach those to her twenty younger brothers and sisters.
4. The concept that I found most useful from lesson 8-1 was the distance formula. Typically the distance formula is used, on a graph, to find the distance between two points. However, the distance formula is used in our every day lives quite frequently. It is used to find the distance between two cities, which we see on road signs. It is used in football games by saying how many yards it will take for the football to get from the line of scrimage to the first down marker. Some things we use the distance formula for, we don't even realize that we are using it. The distance formula is something that you can use for the rest of your life, evn when you are out of your math class!
5. y = (-1/2)x + 3
April 29, 2011
1. Science and mathematics are so closely related that they could easily be intertwined. Many of the great mathematicians could be great in the science field as well and that could be said for many scientists as well about the field of mathematics. One example of this is Johann Kepler.
2.
3. The most difficult problem that I faced in chapter 8 was in section 8-7( #42 on page 472).
2x - y + 3z = 6
x + 2y - z = 8
2y + z = 1
First, I decided to get rid of the z's so I chose the bottom two equations.
x + 2y - z = 8
2y + z = 1
x + 4y = 9
Then, I chose the top and the bottom (multiplied by -3) to eliminate z as well.
2x - y + 3z = 6
-6y - 3z =-3
2x - 7y = 3
I took my two result equations and eliminated x. to do this I multiplied my first result equation by -2. Then I solved for y.
-2x - 8y = -18
2x - 7y = 3
-13y=-15
y=(15/13)
May 13, 2011
1. The Greeks believed that geometry best expressed mathematical science. The Greeks felt that rulers were inadequate for making constructions and they only allowed two instruments: a compass and a straight edge. The straight edge could have absolutely no markins on it and they could not line up points by eye. There were three constructions that the Greeks could not construct with these tools and are known as the three famous problems of aniquity.
-To trisect an arbitrary angle.
-To construct the length of the edge of a cube having twice the volume of a given cube.
-To construct a square having the same area as that of a given circle.
I find the idea of geometry being such a powerful mathematical represetation of science to be very interesting. I myself have been an algebra person because all you need to know is how to set up a formula and then it's easy to solve it. Lately, however, my feeling toward geometry have changed exponentially and I have had a lot of fun working on constructions in my math class (We did have to use a straight edge with markings on it though). I undertand how the Greeks could feel so strongly toward geometry, and in time I might feel the same way.
2. Geometrical shapes are a large part of our world whether or not we always notice them. They are seen all throughout the day and we just don't think about them. The shape that most people see andthink about is the octagon. The octagon is on the famous 'STOP' sign that some of us refuse to abide by. Imagine how wierd the 'STOP' sign would look if it was a triangle. Speaking of triangles, if you live in a house the chances of your roof being held above your house rather than in it, come from our triangular friends "rafters".
If you have ever ordered a pizza, which I have definately ordered my share, then you know that it comes in a nice square pizza box. Who knew that a square would have such a delicious treat stowed away inside of it? Rectangles are the friends to many sports fans that play and/or watch sports. Many of the sports in the world are played on rectangular fields or courts. The pentagon, afive sided shape is also the headquarters of the United States Department of Defense. The building is actually in the shape of a pentagon.
3. Euclid is known as the "Father of Geometry" because he, like Thales, Pathagoras, Plato and more, found it easier to calculate things with geometry. The Greek number system was not a positional system which means that there were no zero's and there were only whole numbers which made it very difficult to make calculations. Euclid decided to use a compass and a straight edge to do calculations in geometrical form and these techniques later became known as Euclidean Constructions. He was the author The Elements, which covered plane geometry, arithmetic and number theory, irrational numbers and solid geometry.
If i could ask Euclid any question, I would ask him how he proved the Pythagorean Theorem. I have seen many different proofs and I, myself, could solve it in many different ways, but I would love to see Euclid solve it. I would imagine that he would come up with the most complex way to solve it and it would probably take a long time as well, but I would be very pleased to his method of solving the Pythagorean Theorem.
4. By definition, a parallelogram is a quadrilateral, meaning four sides, with two pairs of parallel sides. A rhombus is a parallelogram with all sides having equal length. A rhombus has all the criteria to be considered a parallelogram because it has two pairs of parallel sides. However, a parallelogram does have two pairs of parallel sides, like a rhombus, but not all of the sides are equal. If all of the sides were equal, then it would be a rhombus and a parallelogram.